Why is the view header showing triangle count when my mesh is all quads?
Without getting too technical, in 3d, there is no such thing as a true quad. Quads and ngons are mere abstractions to make objects easier to manipulate (imagine modeling a car or face using triangles only). At the very core however, every object is comprised of triangles. Blender and other modeling packages abstract these away but they are there.

You might have seen proof of this if you've ever badly transformed a face for example, if you were to paint all 4 vertices of a plane and move a vertex out of place, you will see evidence that there is supporting geometry somewhere holding that "quad" together. 



To test, just add a plane and check the header again, you will see that it shows 2 tris, this is accurate because a flat plane in its simplest form is two triangles. If you were to cut it diagonally with say the knife tool, it would still show 2 because now it is still 2 triangles.



Now if you were to create an actual triangle, it would show 1.



In a perfect sphere without any actual triangles, say a cube morphed into a sphere (by subdividing and using To Sphere AltShiftS), the amount of triangles will always be twice that of faces, because remember, by default a quad is 2 triangles.

This might not be the case for edges however, that count depends on how uniform your object is, toplogy etc.